How to prove that an exterior angle of a cyclic quadrilateral is equal to its opposite interior angle?

1 Answer
Dec 26, 2017

Use two other theorems to show that this is true.

Explanation:

This uses two other theorems.

$1.$ The opposite angles of a cyclic quad are supplementary.

So x + y =180°

$2.$ Adjacent angles on a straight line are supplementary.

So x_1 +x_2 = 180°

An exterior angle of any shape is formed by extending a side.

Therefore for cyclic quadrilateral with a vertex $X$ lengthened to create an exterior angle and an opposite vertex $Y$ ,

x_1 + x_2 = 180° " " (adj angles on str line)
x_1 +y = 180°" "(opp angles cyclic quad)

$\therefore {x}_{2} = y$

The exterior angle of a cyclic quad is equal to the interior opposite angle.